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1

Maximum number of limit cycles for generalized Kukles differential system

Melki Houdeifa, Makhlouf Amar

Journal of Applied Analysis

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2023

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Vol. 29, nr 1

59--75

EN

We apply the averaging theory of first and second order to a class of generalized polynomial Kukles differential systems, which can bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x, in order to study the maximum number of limit cycles of these systems.

2

Numerical criterion for the duration of non-chaotic transients in ODEs

Szczebiot Ryszard, Kaczyński Roman, Gołdyn Leszek

Acta Mechanica et Automatica

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2022

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Vol. 16, no. 4

388--392

EN

The paper proposes an original numerical criterion for the duration analysis of non-chaotic transients based on the Euclidean norm of a properly defined vector. For this purpose, transient trajectories, prior to their entering a small neighbourhood of the limit cycle, are used. The vector has been defined with its components constituting the lengths of the sections, which connect the origin of the coordinate system with appropriately determined transient trajectory points. The norm of the vector for the analysis of non-chaotic transients has also been applied. As an assessment criterion of transients, the convergence of the norm to small neighbourhood of the limit cycle with the assumed accuracy is used. The paper also provides examples of the application of this criterion to the Van der Pol oscillators in the case of periodic oscillations.

3

Application of natural oscillation values and limit cycles in the analyses of rail vehicles properties

Piechowiak T., Sienicki A.

Pojazdy Szynowe

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2017

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Nr 2

15--29

EN

The article discusses the application of natural oscillation values and limit cycles in the analyses of the properties of rail vehicles on the basis of experiences of the authors. It is presented the exemplary values of natural oscillation obtained with the use of postprocessors: the amplitude-phase graphs, the root lines and stability cards which make the carrying out of analysis performed by the authors easier. It is also shown the application of limit cycles as a function of vehicle speed to determine the riding stability.

PL

Artykuł omawia wykorzystanie wartości drgań własnych i cykli granicznych w analizach właściwości pojazdów szynowych na podstawie doświadczeń autorów. Przedstawiono przykładowe wartości drgań własnych uzyskane z wykorzystaniem postprocesorów: wykresów amplitudowo-fazowych, linii pierwiastkowych i kart stabilności, ułatwiających przeprowadzenie analiz wykonanych przez autorów. Przedstawiono także wykorzystanie cykli granicznych w funkcji prędkości jazdy pojazdu w celu określenia stabilności jazdy.

4

Note on describing function analysis of fractional order nonlinear control systems

Yeroglu C., Tan N.

Control and Cybernetics

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2015

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Vol. 44, no. 2

233--255

EN

This paper presents extensions of some results, obtained for the analysis of classical nonlinear control systems, to the nonlinear fractional order systems. It is shown that the results related to limit cycle prediction using describing function method can be applied to the fractional order plants. The frequency and the amplitude of the limit cycle are used for auto-tuning of the PID controller for nonlinear control systems with fractional order transfer functions. Fractional order control system with parametric uncertainty is also considered for the nonlinear case. On the other hand, a New method is provided for stability margin computation for fractional order nonlinear control system with parametric uncertainty structure using the Nyquist envelopes of the fractional order uncertain plant and the describing function that represents the nonlinearity of the system. Maximum perturbation bounds of the parameters of the fractional order plant are computed. Numerical examples are included to illustrate the methods presented.

5

Classical and generalized invariant solutions of the relaxing hydrodynamics equations

Vladimirov V.

Opuscula Mathematica

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2003

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Vol. 23

81-94

EN

We analyze the set of invariant travelling wave solutions of the relaxing hydrodynamics models and state the conditions that guarantee the existence of periodic solutions and limiting to them soliton-like regimes. We also study the existence of the shock wave invariant solutions. In some special case the periodic invariant solution is shown to give rise to the blow-up regime.

6

Asymptotic behaviour and existence of a limit cycle of cubic autonomous systems

Barakova L.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

559-576

EN

In this paper a 2-dimensional real autonomous system with polynomial right-hand sides of a concrete type is studied. Hopf bifurcation is analysed and existence of a limit cycle is proved. A new formula to determine stability or unstability of this limit cycle is introduced. A positively invariant set, which is globally attractive, is found. Consequently, existence of a stable limit cycle around an unstable critical point is proved and also a sufficient condition for non-existence of a closed trajectory in the phase space is given. Global characteristics of the system are studied. An application in economics to the dynamic version of the neo-keynesian macroeconomic IS-LM model is presented.